abaqus benchmarks manual

Abaqus Benchmarks Manual

Abaqus ID:

Printed on:

Abaqus 6.12Benchmarks Manual

Abaqus

Benchmarks Manual

Abaqus ID:

Printed on:

Legal NoticesCAUTION: This documentation is intended for qualified users who will exercise sound engineering judgment and expertise in the use of the Abaqus

Software. The Abaqus Software is inherently complex, and the examples and procedures in this documentation are not intended to be exhaustive or to apply

to any particular situation. Users are cautioned to satisfy themselves as to the accuracy and results of their analyses.

Dassault Systmes and its subsidiaries, including Dassault Systmes Simulia Corp., shall not be responsible for the accuracy or usefulness of any analysis

performed using the Abaqus Software or the procedures, examples, or explanations in this documentation. Dassault Systmes and its subsidiaries shall not

be responsible for the consequences of any errors or omissions that may appear in this documentation.

The Abaqus Software is available only under license from Dassault Systmes or its subsidiary and may be used or reproduced only in accordance with the

terms of such license. This documentation is subject to the terms and conditions of either the software license agreement signed by the parties, or, absent

such an agreement, the then current software license agreement to which the documentation relates.

This documentation and the software described in this documentation are subject to change without prior notice.

No part of this documentation may be reproduced or distributed in any form without prior written permission of Dassault Systmes or its subsidiary.

The Abaqus Software is a product of Dassault Systmes Simulia Corp., Providence, RI, USA.

Dassault Systmes, 2012

Abaqus, the 3DS logo, SIMULIA, CATIA, and Unified FEA are trademarks or registered trademarks of Dassault Systmes or its subsidiaries in the United

States and/or other countries.

Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerning

trademarks, copyrights, and licenses, see the Legal Notices in the Abaqus 6.12 Installation and Licensing Guide.

Abaqus ID:

Printed on:

Locations

SIMULIA Worldwide Headquarters Rising Sun Mills, 166 Valley Street, Providence, RI 029092499, Tel: +1 401 276 4400,

Fax: +1 401 276 4408, [email protected], http://www.simulia.com

SIMULIA European Headquarters Stationsplein 8-K, 6221 BT Maastricht, The Netherlands, Tel: +31 43 7999 084,

Fax: +31 43 7999 306, [email protected]

Dassault Systmes Centers of Simulation Excellence

United States Fremont, CA, Tel: +1 510 794 5891, [email protected]

West Lafayette, IN, Tel: +1 765 497 1373, [email protected]

Northville, MI, Tel: +1 248 349 4669, [email protected]

Woodbury, MN, Tel: +1 612 424 9044, [email protected]

Mayfield Heights, OH, Tel: +1 216 378 1070, [email protected]

Mason, OH, Tel: +1 513 275 1430, [email protected]

Warwick, RI, Tel: +1 401 739 3637, [email protected]

Lewisville, TX, Tel: +1 972 221 6500, [email protected]

Australia Richmond VIC, Tel: +61 3 9421 2900, [email protected]

Austria Vienna, Tel: +43 1 22 707 200, [email protected]

Benelux Maarssen, The Netherlands, Tel: +31 346 585 710, [email protected]

Canada Toronto, ON, Tel: +1 416 402 2219, [email protected]

China Beijing, P. R. China, Tel: +8610 6536 2288, [email protected]

Shanghai, P. R. China, Tel: +8621 3856 8000, [email protected]

Finland Espoo, Tel: +358 40 902 2973, [email protected]

France Velizy Villacoublay Cedex, Tel: +33 1 61 62 72 72, [email protected]

Germany Aachen, Tel: +49 241 474 01 0, [email protected]

Munich, Tel: +49 89 543 48 77 0, [email protected]

India Chennai, Tamil Nadu, Tel: +91 44 43443000, [email protected]

Italy Lainate MI, Tel: +39 02 3343061, [email protected]

Japan Tokyo, Tel: +81 3 5442 6302, [email protected]

Osaka, Tel: +81 6 7730 2703, [email protected]

Korea Mapo-Gu, Seoul, Tel: +82 2 785 6707/8, [email protected]

Latin America Puerto Madero, Buenos Aires, Tel: +54 11 4312 8700, [email protected]

Scandinavia Stockholm, Sweden, Tel: +46 8 68430450, [email protected]

United Kingdom Warrington, Tel: +44 1925 830900, [email protected]

Authorized Support Centers

Argentina SMARTtech Sudamerica SRL, Buenos Aires, Tel: +54 11 4717 2717

KB Engineering, Buenos Aires, Tel: +54 11 4326 7542

Solaer Ingeniera, Buenos Aires, Tel: +54 221 489 1738

Brazil SMARTtech Mecnica, Sao Paulo-SP, Tel: +55 11 3168 3388

Czech & Slovak Republics Synerma s. r. o., Psry, Prague-West, Tel: +420 603 145 769, [email protected]

Greece 3 Dimensional Data Systems, Crete, Tel: +30 2821040012, [email protected]

Israel ADCOM, Givataim, Tel: +972 3 7325311, [email protected]

Malaysia WorleyParsons Services Sdn. Bhd., Kuala Lumpur, Tel: +603 2039 9000, [email protected]

Mexico Kimeca.NET SA de CV, Mexico, Tel: +52 55 2459 2635

New Zealand Matrix Applied Computing Ltd., Auckland, Tel: +64 9 623 1223, [email protected]

Poland BudSoft Sp. z o.o., Pozna, Tel: +48 61 8508 466, [email protected]

Russia, Belarus & Ukraine TESIS Ltd., Moscow, Tel: +7 495 612 44 22, [email protected]

Singapore WorleyParsons Pte Ltd., Singapore, Tel: +65 6735 8444, [email protected]

South Africa Finite Element Analysis Services (Pty) Ltd., Parklands, Tel: +27 21 556 6462, [email protected]

Spain & Portugal Principia Ingenieros Consultores, S.A., Madrid, Tel: +34 91 209 1482, [email protected]

Abaqus ID:

Printed on:

Taiwan Simutech Solution Corporation, Taipei, R.O.C., Tel: +886 2 2507 9550, [email protected]

Thailand WorleyParsons Pte Ltd., Singapore, Tel: +65 6735 8444, [email protected]

Turkey A-Ztech Ltd., Istanbul, Tel: +90 216 361 8850, [email protected]

Complete contact information is available at http://www.simulia.com/locations/locations.html.

Abaqus ID:

Printed on:

Preface

This section lists various resources that are available for help with using Abaqus Unified FEA software.

Support

Both technical engineering support (for problems with creating a model or performing an analysis) and

systems support (for installation, licensing, and hardware-related problems) for Abaqus are offered through

a network of local support offices. Regional contact information is listed in the front of each Abaqus manual

and is accessible from the Locations page at www.simulia.com.

Support for SIMULIA products

SIMULIA provides a knowledge database of answers and solutions to questions that we have answered,

as well as guidelines on how to use Abaqus, SIMULIA Scenario Definition, Isight, and other SIMULIA

products. You can also submit new requests for support. All support incidents are tracked. If you contact

us by means outside the system to discuss an existing support problem and you know the incident or support

request number, please mention it so that we can query the database to see what the latest action has been.

Many questions about Abaqus can also be answered by visiting the Products page and the Supportpage at www.simulia.com.

Anonymous ftp site

To facilitate data transfer with SIMULIA, an anonymous ftp account is available at ftp.simulia.com.

Login as user anonymous, and type your e-mail address as your password. Contact support before placing

files on the site.

Training

All offices and representatives offer regularly scheduled public training classes. The courses are offered in

a traditional classroom form and via the Web. We also provide training seminars at customer sites. All

training classes and seminars include workshops to provide as much practical experience with Abaqus as

possible. For a schedule and descriptions of available classes, see www.simulia.com or call your local office

or representative.

Feedback

We welcome any suggestions for improvements to Abaqus software, the support program, or documentation.

We will ensure that any enhancement requests you make are considered for future releases. If you wish to

make a suggestion about the service or products, refer to www.simulia.com. Complaints should be made by

contacting your local office or through www.simulia.com by visiting the Quality Assurance section of theSupport page.

Abaqus ID:

Printed on:

CONTENTS

Contents

1. Analysis Tests

Static stress/displacement analysis

Beam/gap example 1.1.1

Analysis of an anisotropic layered plate 1.1.2

Composite shells in cylindrical bending 1.1.3

Thick composite cylinder subjected to internal pressure 1.1.4

Uniform collapse of straight and curved pipe segments 1.1.5

Snap-through of a shallow, cylindrical roof under a point load 1.1.6

Pressurized rubber disc 1.1.7

Uniaxial stretching of an elastic sheet with a circular hole 1.1.8

Necking of a round tensile bar 1.1.9

Concrete slump test 1.1.10

The Hertz contact problem 1.1.11

Crushing of a pipe 1.1.12

Buckling analysis

Buckling analysis of beams 1.2.1

Buckling of a ring in a plane under external pressure 1.2.2

Buckling of a cylindrical shell under uniform axial pressure 1.2.3

Buckling of a simply supported square plate 1.2.4

Lateral buckling of an L-bracket 1.2.5

Buckling of a column with general contact 1.2.6

Dynamic stress/displacement analysis

Subspace dynamic analysis of a cantilever beam 1.3.1

Double cantilever elastic beam under point load 1.3.2

Explosively loaded cylindrical panel 1.3.3

Free ring under initial velocity: comparison of rate-independent and rate-dependent

plasticity 1.3.4

Large rotation of a one degree of freedom system 1.3.5

Motion of a rigid body in Abaqus/Standard 1.3.6

Rigid body dynamics with Abaqus/Explicit 1.3.7

Revolute MPC verification: rotation of a crank 1.3.8

Pipe whip simulation 1.3.9

Impact of a copper rod 1.3.10

Frictional braking of a rotating rigid body 1.3.11

Compression of cylindrical shells with general contact 1.3.12

Steady-state slip of a belt drive 1.3.13

i

Abaqus ID:bmk-toc

Printed on: Fri February 3 -- 10:30:54 2012

CONTENTS

Crash simulation of a motor vehicle 1.3.14

Truss impact on a rigid wall 1.3.15

Plate penetration by a projectile 1.3.16

Oblique shock reflections 1.3.17

Mode-based dynamic analysis

Free vibrations of a spherical shell 1.4.1

Eigenvalue analysis of a beam under various end constraints and loadings 1.4.2

Vibration of a cable under tension 1.4.3

Free and forced vibrations with damping 1.4.4

Verification of Rayleigh damping options with direct integration and modal

superposition 1.4.5

Eigenvalue analysis of a cantilever plate 1.4.6

Vibration of a rotating cantilever plate 1.4.7

Response spectrum analysis of a simply supported beam 1.4.8

Linear analysis of a rod under dynamic loading 1.4.9

Random response to jet noise excitation 1.4.10

Random response of a cantilever subjected to base motion 1.4.11

Double cantilever subjected to multiple base motions 1.4.12

Analysis of a cantilever subject to earthquake motion 1.4.13

Residual modes for modal response analysis 1.4.14

Steady-state transport analysis

Steady-state transport analysis 1.5.1

Steady-state spinning of a disk in contact with a foundation 1.5.2

Heat transfer and thermal-stress analysis

Convection and diffusion of a temperature pulse 1.6.1

Freezing of a square solid: the two-dimensional Stefan problem 1.6.2

Coupled temperature-displacement analysis: one-dimensional gap conductance and

radiation 1.6.3

Quenching of an infinite plate 1.6.4

Two-dimensional elemental cavity radiation viewfactor calculations 1.6.5

Axisymmetric elemental cavity radiation viewfactor calculations 1.6.6

Three-dimensional elemental cavity radiation viewfactor calculations 1.6.7

Radiation analysis of a plane finned surface 1.6.8

Eulerian analysis

Eulerian analysis of a collapsing water column 1.7.1

Deflection of an elastic dam under water pressure 1.7.2

Electromagnetic analysis

Eigenvalue analysis of a piezoelectric cube with various electrode configurations 1.8.1

Modal dynamic analysis for piezoelectric materials 1.8.2

ii

Abaqus ID:bmk-toc


CONTENTS

Steady-state dynamic analysis for piezoelectric materials 1.8.3

TEAM 2: Eddy current simulations of long cylindrical conductors in an oscillating

magnetic field 1.8.4

TEAM 6: Eddy current simulations for spherical conductors in an oscillating magnetic

field 1.8.5

Induction heating of a cylindrical rod by an encircling coil carrying time-harmonic

current 1.8.6

Coupled pore fluid flow and stress analysis

Partially saturated flow in a porous medium 1.9.1

Demand wettability of a porous medium: coupled analysis 1.9.2

Wicking in a partially saturated porous medium 1.9.3

Desaturation in a column of porous material 1.9.4

Mass diffusion analysis

Thermo-mechanical diffusion of hydrogen in a bending beam 1.10.1

Acoustic analysis

A simple coupled acoustic-structural analysis 1.11.1

Analysis of a point-loaded, fluid-filled, spherical shell 1.11.2

Acoustic radiation impedance of a sphere in breathing mode 1.11.3

Acoustic-structural interaction in an infinite acoustic medium 1.11.4

Acoustic-acoustic tie constraint in two dimensions 1.11.5

Acoustic-acoustic tie constraint in three dimensions 1.11.6

A simple steady-state dynamic acoustic analysis 1.11.7

Acoustic analysis of a duct with mean flow 1.11.8

Real exterior acoustic eigenanalysis 1.11.9

Coupled exterior acoustic eigenanalysis 1.11.10

Acoustic scattering from a rigid sphere 1.11.11

Acoustic scattering from an elastic spherical shell 1.11.12

Adaptivity analysis

Indentation with different materials 1.12.1

Wave propagation with different materials 1.12.2

Adaptivity patch test with different materials 1.12.3

Wave propagation in a shock tube 1.12.4

Propagation of a compaction wave in a shock tube 1.12.5

Advection in a rotating frame 1.12.6

Water sloshing in a pitching tank 1.12.7

Abaqus/Aqua analysis

Pull-in of a pipeline lying directly on the seafloor 1.13.1

Near bottom pipeline pull-in and tow 1.13.2

Slender pipe subject to drag: the reed in the wind 1.13.3

iii

Abaqus ID:bmk-toc


CONTENTS

Underwater shock analysis

One-dimensional underwater shock analysis 1.14.1

The submerged sphere problem 1.14.2

The submerged infinite cylinder problem 1.14.3

The one-dimensional cavitation problem 1.14.4

Plate response to a planar exponentially decaying shock wave 1.14.5

Cylindrical shell response to a planar step shock wave 1.14.6

Cylindrical shell response to a planar exponentially decaying shock wave 1.14.7

Spherical shell response to a planar step wave 1.14.8

Spherical shell response to a planar exponentially decaying wave 1.14.9

Spherical shell response to a spherical exponentially decaying wave 1.14.10

Air-backed coupled plate response to a planar exponentially decaying wave 1.14.11

Water-backed coupled plate response to a planar exponentially decaying wave 1.14.12

Coupled cylindrical shell response to a planar step wave 1.14.13

Coupled spherical shell response to a planar step wave 1.14.14

Fluid-filled spherical shell response to a planar step wave 1.14.15

Response of beam elements to a planar wave 1.14.16

Soils analysis

The Terzaghi consolidation problem 1.15.1

Consolidation of a triaxial test specimen 1.15.2

Finite-strain consolidation of a two-dimensional solid 1.15.3

Limit load calculations with granular materials 1.15.4

Finite deformation of an elastic-plastic granular material 1.15.5

The one-dimensional thermal consolidation problem 1.15.6

Consolidation around a cylindrical heat source 1.15.7

Fracture mechanics

Contour integral evaluation: two-dimensional case 1.16.1

Contour integral evaluation: three-dimensional case 1.16.2

Center slant cracked plate under tension 1.16.3

A penny-shaped crack under concentrated forces 1.16.4

Fully plastic J -integral evaluation 1.16.5

Ct-integral evaluation 1.16.6

Nonuniform crack-face loading and J -integrals 1.16.7

Single-edged notched specimen under a thermal load 1.16.8

Substructures

Analysis of a frame using substructures 1.17.1

Design sensitivity analysis

Design sensitivity analysis for cantilever beam 1.18.1

iv

Abaqus ID:bmk-toc


CONTENTS

Sensitivity of the stress concentration factor around a circular hole in a plate under

uniaxial tension 1.18.2

Sensitivity analysis of modified NAFEMS problem 3DNLG-1: Large deflection of

Z-shaped cantilever under an end load 1.18.3

Modeling discontinuities using XFEM

Crack propagation of a single-edge notch simulated using XFEM 1.19.1

Crack propagation in a plate with a hole simulated using XFEM 1.19.2

Crack propagation in a beam under impact loading simulated using XFEM 1.19.3

Dynamic shear failure of a single-edge notch simulated using XFEM 1.19.4

2. Element Tests

Continuum elements

Torsion of a hollow cylinder 2.1.1

Geometrically nonlinear analysis of a cantilever beam 2.1.2

Cantilever beam analyzed with CAXA and SAXA elements 2.1.3

Two-point bending of a pipe due to self weight: CAXA and SAXA elements 2.1.4

Cooks membrane problem 2.1.5

Infinite elements

Wave propagation in an infinite medium 2.2.1

Infinite elements: the Boussinesq and Flamant problems 2.2.2

Infinite elements: circular load on half-space 2.2.3

Spherical cavity in an infinite medium 2.2.4

Structural elements

The barrel vault roof problem 2.3.1

The pinched cylinder problem 2.3.2

The pinched sphere problem 2.3.3

Skew sensitivity of shell elements 2.3.4

Performance of continuum and shell elements for linear analysis of bending problems 2.3.5

Tip in-plane shear load on a cantilevered hook 2.3.6

Analysis of a twisted beam 2.3.7

Twisted ribbon test for shells 2.3.8

Ribbon test for shells with applied moments 2.3.9

Triangular plate-bending on three point supports 2.3.10

Shell elements subjected to uniform thermal loading 2.3.11

Shell bending under a tip load 2.3.12

Variable thickness shells and membranes 2.3.13

Transient response of a shallow spherical cap 2.3.14

Simulation of propeller rotation 2.3.15

v

Abaqus ID:bmk-toc


CONTENTS

Acoustic elements

Acoustic modes of an enclosed cavity 2.4.1

Fluid elements

Fluid filled rubber bladders 2.5.1

Connector elements

Dynamic response of a two degree of freedom system 2.6.1

Linear behavior of spring and dashpot elements 2.6.2

Special-purpose elements

Delamination analysis of laminated composites 2.7.1

3. Material Tests

Elasticity

Viscoelastic rod subjected to constant axial load 3.1.1

Transient thermal loading of a viscoelastic slab 3.1.2

Uniform strain, viscoplastic truss 3.1.3

Fitting of rubber test data 3.1.4

Fitting of elastomeric foam test data 3.1.5

Rubber under uniaxial tension 3.1.6

Anisotropic hyperelastic modeling of arterial layers 3.1.7

Plasticity and creep

Uniformly loaded, elastic-plastic plate 3.2.1

Test of ORNL plasticity theory under biaxial loading 3.2.2

One-way reinforced concrete slab 3.2.3

Triaxial tests on a saturated clay 3.2.4

Uniaxial tests on jointed material 3.2.5

Verification of creep integration 3.2.6

Simple tests on a crushable foam specimen 3.2.7

Simple proportional and nonproportional cyclic tests 3.2.8

Biaxial tests on gray cast iron 3.2.9

Indentation of a crushable foam plate 3.2.10

Notched unreinforced concrete beam under 3-point bending 3.2.11

Mixed-mode failure of a notched unreinforced concrete beam 3.2.12

Slider mechanism with slip-rate-dependent friction 3.2.13

Cylinder under internal pressure 3.2.14

Creep of a thick cylinder under internal pressure 3.2.15

Pressurization of a thick-walled cylinder 3.2.16

Stretching of a plate with a hole 3.2.17

Pressure on infinite geostatic medium 3.2.18

vi

Abaqus ID:bmk-toc


CONTENTS

4. NAFEMS Benchmarks

Overview

NAFEMS benchmarks: overview 4.1.1

Standard benchmarks: linear elastic tests

LE1: Plane stress elementselliptic membrane 4.2.1

LE2: Cylindrical shell bending patch test 4.2.2

LE3: Hemispherical shell with point loads 4.2.3

LE4: Axisymmetric hyperbolic shell under uniform internal pressure 4.2.4

LE5: Z-section cantilever 4.2.5

LE6: Skew plate under normal pressure 4.2.6

LE7: Axisymmetric cylinder/sphere under pressure 4.2.7

LE8: Axisymmetric shell under pressure 4.2.8

LE9: Axisymmetric branched shell under pressure 4.2.9

LE10: Thick plate under pressure 4.2.10

LE11: Solid cylinder/taper/spheretemperature loading 4.2.11

Standard benchmarks: linear thermo-elastic tests

T1: Plane stress elementsmembrane with hot-spot 4.3.1

T2: One-dimensional heat transfer with radiation 4.3.2

T3: One-dimensional transient heat transfer 4.3.3

T4: Two-dimensional heat transfer with convection 4.3.4

Standard benchmarks: free vibration tests

FV2: Pin-ended double cross: in-plane vibration 4.4.1

FV4: Cantilever with off-center point masses 4.4.2

FV12: Free thin square plate 4.4.3

FV15: Clamped thin rhombic plate 4.4.4

FV16: Cantilevered thin square plate 4.4.5

FV22: Clamped thick rhombic plate 4.4.6

FV32: Cantilevered tapered membrane 4.4.7

FV41: Free cylinder: axisymmetric vibration 4.4.8

FV42: Thick hollow sphere: uniform radial vibration 4.4.9

FV52: Simply supported solid square plate 4.4.10

Proposed forced vibration benchmarks

Test 5: Deep simply supported beam: frequency extraction 4.5.1

Test 5H: Deep simply supported beam: harmonic forced vibration 4.5.2

Test 5T: Deep simply supported beam: transient forced vibration 4.5.3

Test 5R: Deep simply supported beam: random forced vibration 4.5.4

Test 13: Simply supported thin square plate: frequency extraction 4.5.5

Test 13H: Simply supported thin square plate: harmonic forced vibration 4.5.6

vii

Abaqus ID:bmk-toc


CONTENTS

Test 13T: Simply supported thin square plate: transient forced vibration 4.5.7

Test 13R: Simply supported thin square plate: random forced vibration 4.5.8

Test 21: Simply supported thick square plate: frequency extraction 4.5.9

Test 21H: Simply supported thick square plate: harmonic forced vibration 4.5.10

Test 21T: Simply supported thick square plate: transient forced vibration 4.5.11

Test 21R: Simply supported thick square plate: random forced vibration 4.5.12

Proposed nonlinear benchmarks

NL1: Prescribed biaxial strain history, plane strain 4.6.1

NL2: Axisymmetric thick cylinder 4.6.2

NL3: Hardening with two variables under load control 4.6.3

NL4: Snap-back under displacement control 4.6.4

NL5: Straight cantilever with end moment 4.6.5

NL6: Straight cantilever with axial end point load 4.6.6

NL7: Lees frame buckling problem 4.6.7

Two-dimensional test cases in linear elastic fracture mechanics

Test 1.1: Center cracked plate in tension 4.7.1

Test 1.2: Center cracked plate with thermal load 4.7.2

Test 2.1: Single edge cracked plate in tension 4.7.3

Test 3: Angle crack embedded in a plate 4.7.4

Test 4: Cracks at a hole in a plate 4.7.5

Test 5: Axisymmetric crack in a bar 4.7.6

Test 6: Compact tension specimen 4.7.7

Test 7.1: T-joint weld attachment 4.7.8

Test 8.1: V-notch specimen in tension 4.7.9

Fundamental tests of creep behavior

Test 1A: 2-D plane stress uniaxial load, secondary creep 4.8.1

Test 1B: 2-D plane stress uniaxial displacement, secondary creep 4.8.2

Test 2A: 2-D plane stress biaxial load, secondary creep 4.8.3

Test 2B: 2-D plane stress biaxial displacement, secondary creep 4.8.4

Test 3A: 2-D plane stress biaxial (negative) load, secondary creep 4.8.5

Test 3B: 2-D plane stress biaxial (negative) displacement, secondary creep 4.8.6

Test 4A: 2-D plane stress biaxial (double) load, secondary creep 4.8.7

Test 4B: 2-D plane stress biaxial (double) displacement, secondary creep 4.8.8

Test 4C: 2-D plane stress shear loading, secondary creep 4.8.9

Test 5A: 2-D plane strain biaxial load, secondary creep 4.8.10

Test 5B: 2-D plane strain biaxial displacement, secondary creep 4.8.11

Test 6A: 3-D triaxial load, secondary creep 4.8.12

Test 6B: 3-D triaxial displacement, secondary creep 4.8.13

Test 7: Axisymmetric pressurized cylinder, secondary creep 4.8.14

Test 8A: 2-D plane stress uniaxial load, primary creep 4.8.15

viii

Abaqus ID:bmk-toc


CONTENTS

Test 8B: 2-D plane stress uniaxial displacement, primary creep 4.8.16

Test 8C: 2-D plane stress stepped load, primary creep 4.8.17

Test 9A: 2-D plane stress biaxial load, primary creep 4.8.18

Test 9B: 2-D plane stress biaxial displacement, primary creep 4.8.19

Test 9C: 2-D plane stress biaxial stepped load, primary creep 4.8.20

Test 10A: 2-D plane stress biaxial (negative) load, primary creep 4.8.21

Test 10B: 2-D plane stress biaxial (negative) displacement, primary creep 4.8.22

Test 10C: 2-D plane stress biaxial (negative) stepped load, primary creep 4.8.23

Test 11: 3-D triaxial load, primary creep 4.8.24

Test 12A: 2-D plane stress uniaxial load, primary-secondary creep 4.8.25

Test 12B: 2-D plane stress uniaxial displacement, primary-secondary creep 4.8.26

Test 12C: 2-D plane stress stepped load, primary-secondary creep 4.8.27

Composite tests

R0031(1): Laminated strip under three-point bending 4.9.1

R0031(2): Wrapped thick cylinder under pressure and thermal loading 4.9.2

R0031(3): Three-layer sandwich shell under normal pressure loading 4.9.3

Geometric nonlinear tests

3DNLG-1: Elastic large deflection response of a Z-shaped cantilever under an end load 4.10.1

3DNLG-2: Elastic large deflection response of a pear-shaped cylinder under end

shortening 4.10.2

3DNLG-3: Elastic lateral buckling of a right angle frame under in-plane end moments 4.10.3

3DNLG-4: Lateral torsional buckling of an elastic cantilever subjected to a transverse

end load 4.10.4

3DNLG-5: Large deflection of a curved elastic cantilever under transverse end load 4.10.5

3DNLG-6: Buckling of a flat plate when subjected to in-plane shear 4.10.6

3DNLG-7: Elastic large deflection response of a hinged spherical shell under pressure

loading 4.10.7

3DNLG-8: Collapse of a straight pipe segment under pure bending 4.10.8

3DNLG-9: Large elastic deflection of a pinched hemispherical shell 4.10.9

3DNLG-10: Elastic-plastic behavior of a stiffened cylindrical panel under compressive

end load 4.10.10

ix

Abaqus ID:bmk-toc


Abaqus ID:

Printed on:

INTRODUCTION

1.0 INTRODUCTION

This is the Benchmarks Manual for Abaqus. It contains benchmark problems (including the NAFEMS suite

of test problems) and standard analyses used to evaluate the performance of Abaqus. The tests in this manual

are multiple element tests of simple geometries or simplified versions of real problems.

In addition to the Benchmarks Manual there are two other manuals that contain worked problems. The

Abaqus Example ProblemsManual contains many solved examples that test the codewith the type of problems

users are likely to solve. Many of these problems are quite difficult and test a combination of capabilities in the

code. The Abaqus Verification Manual contains a large number of examples that are intended as elementary

verification of the basic modeling capabilities in Abaqus.

The qualification process for new Abaqus releases includes running and verifying results for all problems

in the Abaqus Example Problems Manual, the Abaqus Benchmarks Manual, and the Abaqus Verification

Manual.

All input files referred to in the manuals are included with the Abaqus release in compressed archive

files. The abaqus fetch utility is used to extract these input files for use. For example, to fetch input file

barrelvault_s8r5_reg22.inp, type

abaqus fetch job=barrelvault_s8r5_reg22.inp

Parametric study script (.psf) and user subroutine (.f) files can be fetched in the same manner. All files for

a particular problem can be obtained by leaving off the file extension. The abaqus fetch utility is explained

in detail in Fetching sample input files, Section 3.2.14 of the Abaqus Analysis Users Manual.

It is sometimes useful to search the input files. The findkeyword utility is used to locate input files

that contain user-specified input. This utility is defined in Querying the keyword/problem database,

Section 3.2.13 of the Abaqus Analysis Users Manual.

1.01

Abaqus ID:

Printed on:

ANALYSIS TESTS

1. Analysis Tests

Static stress/displacement analysis, Section 1.1

Buckling analysis, Section 1.2

Dynamic stress/displacement analysis, Section 1.3

Mode-based dynamic analysis, Section 1.4

Steady-state transport analysis, Section 1.5

Heat transfer and thermal-stress analysis, Section 1.6

Eulerian analysis, Section 1.7

Electromagnetic analysis, Section 1.8

Coupled pore fluid flow and stress analysis, Section 1.9

Mass diffusion analysis, Section 1.10

Acoustic analysis, Section 1.11

Adaptivity analysis, Section 1.12

Abaqus/Aqua analysis, Section 1.13

Underwater shock analysis, Section 1.14

Soils analysis, Section 1.15

Fracture mechanics, Section 1.16

Substructures, Section 1.17

Design sensitivity analysis, Section 1.18

Modeling discontinuities using XFEM, Section 1.19

Abaqus ID:

Printed on:

STATIC STRESS/DISPLACEMENT ANALYSIS

1.1 Static stress/displacement analysis

Beam/gap example, Section 1.1.1

Analysis of an anisotropic layered plate, Section 1.1.2

Composite shells in cylindrical bending, Section 1.1.3

Thick composite cylinder subjected to internal pressure, Section 1.1.4

Uniform collapse of straight and curved pipe segments, Section 1.1.5

Snap-through of a shallow, cylindrical roof under a point load, Section 1.1.6

Pressurized rubber disc, Section 1.1.7

Uniaxial stretching of an elastic sheet with a circular hole, Section 1.1.8

Necking of a round tensile bar, Section 1.1.9

Concrete slump test, Section 1.1.10

The Hertz contact problem, Section 1.1.11

Crushing of a pipe, Section 1.1.12

1.11

Abaqus ID:

Printed on:

BEAM/GAP EXAMPLE

1.1.1 BEAM/GAP EXAMPLE

Product: Abaqus/Standard

The purpose of this example is to verify the performance of a gap element in a simple case. Three parallel

cantilever beams are initially separate but have possible contact points in five locations, as shown in

Figure 1.1.11. A pair of pinching loads is applied, as shown. Only small displacements are considered,

so each beam responds in pure bending. The problem is entirely linear, except for the switching contact

conditions.

The sequence of events is readily imagined:

1. The top and bottom beams bend as the pinching forces are applied, and the first contact occurs when

the tip of the top beam hits the tip of the middle beam (gap 3 closes). Up to this point the problem is

symmetric about the middle beam, but it now loses that symmetry.

2. Subsequent to this initial contact, the top and middle beams bend down and the bottom beam continues

to bend up until contact occurs at gap 5.

3. As the load continues to increase, gap 2 closes.

4. Next, gap 3 opens as the support provided to the top beam by gap 2 causes the outboard part of the

top beam to reverse its direction of rotation. At this point (when gap 3 opens), the solution becomes

symmetric about the middle beam once again.

5. Finally, as the pinching loads increase further, gaps 1 and 4 also close. From this point on the contact

conditions do not switch, no matter how much more load is applied.

Problem description

Each cantilever is modeled using five cubic beam elements of type B23. Initially all gaps are open, with

an initial gap clearance of 0.01. The pinching loads are increased monotonically from 0 to 200. The

beam lengths, modulus, and cross-section are shown in Figure 1.1.11. (The units of dimension and

force are consistent but not physical.)

The loads are applied in 10 equal increments, with the increment size given directly by using the

DIRECT parameter on the *STATIC option.

Results and discussion

The solution is summarized in Table 1.1.11.

Input file

beamgap.inp Input data for this problem.

1.1.11

Abaqus ID:

Printed on:

BEAM/GAP EXAMPLE

Table 1.1.11 Beam/gap example: solution summary.

Pinching Force in gapIncrementforce, P 1 2 3 4 5

1 20 Open 6.5 0.732 Open 7.97

2 40 Open 18.3 Open Open 18.3





7 140 10.7 68.6 Open 10.7 68.6

8 160 31.6 75.9 Open 31.6 75.9

9 180 52.5 83.2 Open 52.5 83.2

10 200 73.4 90.4 Open 73.4 90.4

P

(1)

(4)

(2)

(5)

(3)

P

Material properties:

Young's modulus

Beam section data:

hexagonal, circumscribing radius = 0.5wall thickness = 0.1

= 108 force/length2

10 10 10 10 10

Figure 1.1.11 Beam/gap example.

1.1.12

Abaqus ID:

Printed on:

ANISOTROPIC COMPOSITE SHELLS

1.1.2 ANALYSIS OF AN ANISOTROPIC LAYERED PLATE


This example illustrates the use of the *ORIENTATION option (Orientations, Section 2.2.5 of the Abaqus

Analysis UsersManual) in the analysis of multilayered, laminated, composite shells. The problem considered

is the linear analysis of a flat plate made from two layers oriented at 45, subjected to a uniform pressure

loading. The example verifies simple laminated composite plate analysis. The Abaqus results are compared

with the analytical solution given in Spilker et al. (1976). The cross-section is not balanced, so the response

includes membrane-bending coupling. Composite failure measures are defined for the plane stress orthotropic

material.

Problem description

The structure is a two-layer, composite, orthotropic, square plate that is simply supported on its edges.

The layers are oriented at 45 with respect to the plate edges. Figure 1.1.21 shows the loading and

the plate dimensions. Each layer has the following material properties:

276 GPa (40 106 lb/in2 )

6.9 GPa (106 lb/in2 )

3.4 GPa (0.5 106 lb/in2 )

0.25

These properties are specified directly in the *ELASTIC, TYPE=LAMINA option (Linear elastic

behavior, Section 22.2.1 of the Abaqus Analysis Users Manual), which is provided for defining linear

elastic behavior for a lamina under plane stress conditions. More general orthotropic properties (for

solid continuum elements) can be specified with the *ELASTIC, TYPE=ORTHOTROPIC option.

In this example the plate is considered to be at an arbitrary angle to the global axis system to make

use of the *ORIENTATION option for illustration purposes. The plate is shown in Figure 1.1.22.

The boundary conditions require that displacements that are transverse and normal to the shell

edges are fixed, but motions that are parallel to the edges are permitted. The *TRANSFORM option

(Transformed coordinate systems, Section 2.1.5 of the Abaqus Analysis Users Manual) has been used

to define a convenient set of local displacement degrees of freedom so that the boundary conditions and

the output of nodal variables can be interpreted more easily.

The *ORIENTATION option is used to define the direction of the layers. The rotation of the material

axes of the layers with respect to the standard directions used by Abaqus for stress and strain components

in shells is defined on data lines in four of the models used and, again for illustration purposes, by means

of user subroutine ORIENT in four other models. The section is not balanced since it has only two layers

in different orientations, which results in membrane-bending coupling. The motion does not exhibit

symmetry for the same reason, and the entire shell must be modeled.

An alternative means of defining the layer orientation is to use the *ORIENTATION option to

define the orientation of the section and then to define the in-plane angle of rotation relative to the

1.1.21

Abaqus ID:

Printed on:


section orientation directly with the layer data following the *SHELL SECTION or *SHELLGENERAL

SECTION option. In this case the section force and section strain are calculated in the section orientation

directions (rather than the default shell directions).

Three types of models are used. One is an 8 8 mesh of S9R5 elements, which are shell elements

that allow transverse shear along lines in the element. However, the analytical solution of Spilker

et al. uses thin shell theory, which neglects transverse shear effects. We have, therefore, introduced

an artificially high transverse shear stiffness in this model by using the *TRANSVERSE SHEAR

STIFFNESS option.

The second type of model is a 16 16 mesh of triangular shells; models for both S3R and SC6R

elements are provided. These elements are general-purpose shell elements that allow transverse shear

deformation. An artificially high transverse shear stiffness is introduced by using the *TRANSVERSE

SHEAR STIFFNESS option. No mesh convergence studies have been performed, but finer meshes

should improve accuracy since these elements use a constant bending strain approximation.

The third type of model is made up of STRI65 shell elements, which are also based on the discrete

Kirchhoff theory. An 8 8 mesh is used.

Failure measures

To demonstrate the use of composite failure measures (Plane stress orthotropic failure measures,

Section 22.2.3 of the Abaqus Analysis Users Manual), limit stresses are defined with the *FAIL

STRESS option. The stress-based failure criteria are defined as follows:

(Psi) (Psi) (Psi) (Psi) S (Psi)

60.0 104 24.0 104 1.0 104 3.0 104 2.0 104 0.0

Printed failure indices are requested for maximum stress theory (MSTRS) and Tsai-Hill theory (TSAIH).

All failure measures are written to the results file (CFAILURE).


Table 1.1.21 summarizes the results by comparing displacement and moment values to the analytical

solution. It is clear by the results presented in the table that all models give good results, with the second-

order models providing higher accuracy than the first-order S3R model, as would be expected.

Figure 1.1.23 shows the failure surface for Tsai-Hill theory (i.e., those stress values

that, for a given , yield a failure index 1.0), along with the stress state at each section point in the

center of the plate. Only section point 6 has a stress state outside the failure surface ( 1.0).

Input files

anisoplate_s3r_orient.inp S3R element model with the orientation for the material

defined with *ORIENTATION.

anisoplate_s3r_usr_orient.inp S3R element model with the orientation for the material

defined in user subroutine ORIENT.

1.1.22

Abaqus ID:

Printed on:


anisoplate_s3r_usr_orient.f User subroutine ORIENT used in

anisoplate_s3r_usr_orient.inp.

anisoplate_sc6r_orient.inp SC6R element model with the orientation for the material

defined with *ORIENTATION.

anisoplate_sc6r_usr_orient.inp SC6R element model with the orientation for the material

defined in user subroutine ORIENT.

anisoplate_sc6r_orient_gensect.inp SC6R model with the orientation for the shell section

defined with *ORIENTATION and the orientation for the

material defined by an angle on the data lines for *SHELL

GENERAL SECTION.

anisoplate_sc6r_usr_orient.f User subroutine ORIENT used in

anisoplate_sc6r_usr_orient.inp.

anisoplate_s9r5_orient.inp S9R5 model with the orientation for the material defined

with *ORIENTATION.

anisoplate_s9r5_usr_orient.inp S9R5 model with the orientation for the material defined

in user subroutine ORIENT.

anisoplate_s9r5_usr_orient.f User subroutine ORIENT used in

anisoplate_s9r5_usr_orient.inp.

anisoplate_s9r5_orient_sect.inp S9R5 model with the orientation for the shell section

defined with *ORIENTATION and the orientation for

the material defined by an angle on the data lines for

*SHELL SECTION.

anisoplate_s9r5_orient_gensect.inp S9R5 model with the orientation for the shell section

defined with *ORIENTATION and the orientation for

the material defined by an angle on the data lines for

*SHELL GENERAL SECTION.

anisoplate_stri65_orient.inp STRI65 element model with the orientation for the

material defined with *ORIENTATION.

anisoplate_stri65_usr_orient.inp STRI65 element model with the orientation for the

material defined in user subroutine ORIENT.

anisoplate_stri65_usr_orient.f User subroutine ORIENT used in

anisoplate_stri65_usr_orient.inp.

Reference

Spilker, R. L., S. Verbiese, O. Orringer, S. E. French, E. A. Witmer, and A. Harris, Use of the

Hybrid-Stress Finite-Element Model for the Static and Dynamic Analysis of Multilayer Composite

Plates and Shells, Report for the Army Materials and Mechanics Research Center, Watertown,

MA, 1976.

1.1.23

Abaqus ID:

Printed on:


Table 1.1.21 Results for pressure loading of anisotropic plate.

Element In-plane disp. at Normal disp. at Moment, ortype center of plate at center of plate

(mm) (mm) (N-mm)

Analytical 0.3762 23.25 42.05

S3R 0.3724 22.86 40.54

SC6R 0.3724 22.84 40.54

STRI65 0.3760 23.24 42.28

S9R5 0.3752 23.25 42.23

1.1.24

Abaqus ID:

Printed on:


h

z

y

Uniform pressure, p

a

x

b

Geometric properties:

Loading:

a = b = 254 mm (10 in)h = 5.08 mm (0.2 in)

p = 689.4 kPa (100 lb/in2)

Figure 1.1.21 Geometry and loading for flat plate.

n = (0.40825, -0.40825, 0.81650)

x

z

y

Figure 1.1.22 Orientation of plate in space.

1.1.25

Abaqus ID:

Printed on:


-4 -2 0 2 4 6 8

11 stress (*10**5)

-4

-2

0

2

22 stress

(*10**4)

LINE VARIABLE SCALE FACTOR

1 section pt. 1 +1.00E+00






-4 -2 0 2 4 6 8

(*10**5)

-4

-2

0

2

(*10**4)


1 +1.00E+00

2 +1.00E+00

-4 -2 0 2 4 6 8

(*10**5)

-4

-2

0

2

(*10**4)


1 +1.00E+00

2 +1.00E+00

3 +1.00E+00

4 +1.00E+00

5 +1.00E+00

6 +1.00E+00

1

2

3

4

5

6

-4 -2 0 2 4 6 8

(*10**5)

-4

-2

0

2

(*10**4)


1 +1.00E+00

2 +1.00E+00

Tsai-Hill failure surface

Figure 1.1.23 The stress state at each section point in thecenter of the plate, plotted with the Tsai-Hill failure surface.

Note that section point 6 has failed.

1.1.26

Abaqus ID:

Printed on:

COMPOSITE SHELLS

1.1.3 COMPOSITE SHELLS IN CYLINDRICAL BENDING

Products: Abaqus/Standard Abaqus/Explicit

This example provides verification of the transverse shear stress calculations in Abaqus for multilayer

composite shells and demonstrates the use of the plane stress orthotropic failure measures. A discussion

of the transverse shear stresses obtained by composite solids in Abaqus/Standard is also included. The

problem consists of a two- or three-layer plate subjected to a sinusoidal distributed load, as described by

Pagano (1969). The resulting transverse shear and axial stresses through the thickness of the plate are

compared to two existing analytical solutions by Pagano (1969). The first solution is derived from classical

laminated plate theory (CPT), while the second is an exact solution from linear elasticity theory.

Problem description

A schematic of the model is shown in Figure 1.1.31. The structure is a composite plate composed of

orthotropic layers of equal thickness. It is simply supported at its ends and bounded along its edges to

impose plane strain conditions in the y-direction. Each layer models a fiber/matrix composite with the

following properties:

172.4 GPa (25 106 lb/in2 )

6.90 GPa (1.0 106 lb/in2 )

3.45 GPa (0.5 106 lb/in2 )

1.38 GPa (0.2 106 lb/in2 )

0.25

where L signifies the direction parallel to the fibers and T signifies the transverse direction. In

Abaqus/Standard two methods are used to specify the lay-up definition for the conventional shell

element model. First, the *SHELL SECTION, COMPOSITE option is used to specify the thickness,

number of integration points, material name, and orientation of each layer. Second, the *SHELL

GENERAL SECTION, COMPOSITE option is used to specify the thickness, material name, and

angle of orientation relative to the section orientation (the default shell directions in this case) for each

layer. In Abaqus/Explicit only the former method is used. The material properties are specified using

the *ELASTIC, TYPE=LAMINA option. The orientation of the fibers in each layer is defined by

an in-plane rotation angle measured relative to the local shell directions or relative to an orientation

definition given with the ORIENTATION parameter on the *SHELL GENERAL SECTION option.

In addition to the methods outlined above, a third method of stacking continuum shell elements

is used to specify the lay-up definition for a composite model. This method can be used effectively to

study localized behavior, since continuum shell elements handle high aspect ratios between the in-plane

dimension and the thickness dimension well.

The lay-up definition for the continuum (solid) element model in Abaqus/Standard is specified using

the *SOLID SECTION, COMPOSITE option. The thickness, material name, and orientation definition

for each layer are specified on the data lines following the *SOLID SECTION option.

1.1.31

Abaqus ID:

Printed on:

COMPOSITE SHELLS

A distributed load with a sinusoidal distribution in space, , is applied to the top

of the composite plate. In Abaqus/Standard the load is applied using user subroutine DLOAD in a static

linear analysis step. In addition, an Abaqus/Standard input file is included that demonstrates the use of the

DCOUP3D element to apply this distributed load. In Abaqus/Explicit the load is applied instantaneously

at time 0.

Two composite plates are analyzed in this example. The first is a two-layer plate with the fibers

oriented parallel and orthogonal to the x-axis in the bottom and top layer, respectively. In the second

plate, which has three layers of equal thickness, the fibers in the outer layers are oriented parallel to the

x-axis, while the fibers in the middle layer are orthogonal to the x-axis. The span-to-thickness ratio of

the plates, , is varied from 4 to 30 in the Abaqus/Standard analysis; in Abaqus/Explicit this ratio

is 4 throughout the analysis.

A 1 10 mesh of second-order S8R shell elements is used to model the plates in Abaqus/Standard.

A 2 10 mesh of first-order S4R shell elements is used to model the plates in Abaqus/Explicit. The S4R,

S8R, and S8RT shell elements are well-suited for modeling thick composite shells since they account

for transverse shear flexibility. Five integration points are specified through the thickness of each layer

with the models that use the *SHELL SECTION option. This provides sufficient data to describe the

stress distributions through the thickness of each layer. For the models that use the *SHELL GENERAL

SECTION option, only three points are available for output. (Since the analysis is linear elastic, three

points are sufficient to determine all fields through the thickness.) The plate with the lowest span-to-

thickness ratio is also analyzed with Abaqus/Standard using a 1 10 mesh of second-order C3D20R

composite solid elements.

To illustrate the stacking capability of continuum shell elements, several meshes are provided for

the two- and three-layer plates with a span-to-thickness ratio of 4. The two-layer plate is modeled with a

2 10 mesh of SC8R elements, each element representing a single layer of the 90/0 composite plate. One

model of the three-layer plate uses a 1 10 mesh of SC8R elements using a single element through the

thickness with a composite section definition. Another model of the three-layer plate uses a 3 10 mesh

of SC8R elements, each element representing a single layer of the 0/90/0 composite plate. Additional

models of the three-layer plate with 6, 12, and 24 elements through the thickness are provided. In these

models each composite layer is modeled with 2, 4, and 8 elements through the thickness, respectively.

Additional input files using SC8R elements are included to illustrate the use of the STACK

DIRECTION parameter to define the stacking and thickness direction independent of the element nodal

connectivity.

Failure measures

The plane stress orthotropic failure measures are defined in Plane stress orthotropic failure measures,

Section 22.2.3 of the Abaqus Analysis Users Manual. To demonstrate their use, let the limit stresses

and limit strains be given as follows (defined with *FAIL STRESS and *FAIL STRAIN):

Stress Values: S

(GPa) 2.07 104 8.28 105 3.45 106 1.03 105 6.89 106

(lb/in2 ) 30.0 12.0 0.5 1.5 1.0

1.1.32

Abaqus ID:

Printed on:

COMPOSITE SHELLS

Strain Values:

17. 102 7. 102 5. 102 1.3 102 11. 102

The scaling factor for the Tsai-Wu coefficient is 0.0. These values are chosen such that failure

occurs under the stress-based failure criteria for the given loading in the two-layer case with 4.


The results for each of the analyses are discussed in the following sections.

Abaqus/Standard results

Figure 1.1.32 shows the maximum z-displacement as a function of the span-to-thickness ratio of the

two- and three-layer plates in a normalized form as

As seen in the figure, the finite element displacements for both the two- and three-layer plates agree well

with the prediction from elasticity theory for a wide range of s values. The CPT results are stiff at low

values of s since shear flexibility is neglected.

For 4, Figure 1.1.33 and Figure 1.1.34 show the transverse shear stress (TSHR13) and the

axial stress (S11) distributions through the plate thickness for the two-layer plate normalized as

and

Figure 1.1.35 and Figure 1.1.36 show the corresponding results for the three-layer plate. It is seen that

the shell element results are much closer to the predictions of CPT than to elasticity theory because of

the assumption of linear stress variation through the thickness in the first-order shear flexible theory used

for elements such as S8R and S4R.

Figure 1.1.37 compares the elasticity solution of the transverse shear distribution for the three-layer

plate to an approximate solution using the output variable SSAVG4. SSAVG4 is the average transverse

shear stress in the local 1-direction. Since SSAVG4 is constant over an element, mesh refinement (in this

case 24 continuum shell elements through the thickness) is typically required to capture the variation of

shear stress through the thickness of the plate.

The output variables CTSHR13 and CTSHR23 offer a more economical alternative to SSAVG4 and

SSAVG5 for estimating shear stress in stacked continuum shells. Figure 1.1.38 and Figure 1.1.39 show

very good agreement between the elasticity solution of the transverse shear distribution for the three-

and two-layer plates to the solution using the output variable CTSHR13 for a 3 10 and 2 10 mesh

of continuum shell elements, respectively. The shear stress computed using CTSHR13 is continuous

1.1.33

Abaqus ID:

Printed on:

COMPOSITE SHELLS

across the continuum shell element interfaces. In addition, while the estimates of the transverse shear

distributions using SSAVG4 and CTSHR13 (shown in Figure 1.1.37 and Figure 1.1.38) are both good,

using CTSHR13 requires a mesh of only 3 continuum shell elements through the thickness, as compared

to 24 elements for SSAVG4.

Figure 1.1.310 compares the transverse shear stress distribution obtained with the solid element

model with the shell element result. The figure shows that the transverse shear stresses predicted by solid

elements do not vanish at the free surfaces of the structure. It also shows that the stress is discontinuous

at layer interfaces. The reason for this is that in the composite solid element, the transverse shear stresses

are obtained directly from the displacement field in contrast to the shell element, where the transverse

shear stresses are obtained from an equilibrium calculation. These deficiencies decrease if the number

of solid elements used in the discretization through the section thickness is increased. Although the

transverse shear stresses are inaccurate, the displacement field and components of stress in the plane of

the layer (not shown here) are in much better agreement with the analytical result. In fact, these results

are somewhat better than the results obtained with the S8R elements. The composite solid elements were

not used to analyze the thinner plates since the solid elements would not have any advantage over plate

elements in that case.

For 10, Figure 1.1.311 and Figure 1.1.312 show that the transverse shear and axial stress

distributions of the finite element resultsalong with the CPT predictionsagree with elasticity theory.

The stress distributions become more accurate with increasing span-to-thickness ratio (as the plate

becomes thinner in comparison to the span).

In Figure 1.1.313 and Figure 1.1.314 the maximum stress theory and Tsai-Wu theory failure

indices are plotted as a function of the normalized distance from the midsurface for the two- and three-

layer cases, respectively. The indices are calculated at the center of the plate for S8R elements with

4. Values of the failure index greater than or equal to 1.0 indicate failure. Discontinuous jumps in the

failure index occur at layer boundaries as a result of the orientation of the material. The strain levels are

well below those required for failure, so no strain-based failure indices are plotted.

Abaqus/Explicit results

The explicit dynamic analysis is run for a sufficiently long time so that a quasi-static state is reachedthat

is, the plates are in steady-state vibration. Since step loadings are applied, static solutions of stresses can

be obtained as half of their vibration amplitudes.

Figure 1.1.315 and Figure 1.1.316 show the transverse shear stress (TSHR13) and the axial stress

(S11) distributions through the plate thickness for the two-layer S4R model normalized as:

and

compared with classical plate theory (CPT) and linear elasticity theory.

1.1.34

Abaqus ID:

Printed on:

COMPOSITE SHELLS

Figure 1.1.317 and Figure 1.1.318 show the corresponding results for the three-layer plate. In

Figure 1.1.319 and Figure 1.1.320, the maximum stress theory and Tsai-Wu theory failure indices are

plotted as a function of the normalized distance from the midsurface for the two- and three-layer cases,

respectively. The indices are calculated at the center of the plate. Values of the failure index greater

than or equal to 1.0 indicate failure. Discontinuous jumps in the failure index occur at layer boundaries

due to the orientation of the material. The strain levels are well below those required for failure, so no

strain-based failure indices are plotted.

Input files

Abaqus/Standard input files

compositeshells_s8r.inp Three-layer plate with 4 using S8R elements.

compositeshells_s8r.f User subroutine defining nonuniform distributed load for

use with compositeshells_s8r.inp.

compositeshells_s8r_gensect.inp Three-layer plate with 4 using S8R elements and

*SHELL GENERAL SECTION.

compositeshells_s8r_gensect.f User subroutine DLOAD used in

compositeshells_s8r_gensect.inp.

compositeshells_s4.inp S4 element model.

compositeshells_s4.f User subroutine DLOAD used in compositeshells_s4.inp.

compositeshells_s4_gensect.inp S4 element model with *SHELL GENERAL SECTION.

compositeshells_s4_gensect.f User subroutine DLOAD used in

compositeshells_s4_gensect.inp.

compositeshells_s4_dcoup3d.inp S4 element model loaded using a DCOUP3D element.

compositeshells_s4r.inp S4R element model.

compositeshells_s4r.f User subroutine DLOAD used in compositeshells_s4r.inp.

compositeshells_s4r_gensect.inp S4R element model with *SHELL GENERAL

SECTION.

compositeshells_s4r_gensect.f User subroutine DLOAD used in

compositeshells_s4r_gensect.inp.

compositeshells_c3d20r.inp C3D20R composite solid element model.

compositeshells_c3d20r.f User subroutine DLOAD used in

compositeshells_c3d20r.inp.

compositeshells_sc8r_stackdir_1.inp SC8R model using STACK DIRECTION=1.



compositeshells_sc8r_gensect.inp SC8R model using *SHELL GENERAL SECTION.

compshell2_std_sc8r_stack_2.inp Two-layer plate with SC8R elements, two elements

stacked through the thickness.

compshell3_std_sc8r_stack_1.inp Three-layer plate with SC8R elements, single element

through the thickness.

1.1.35

Abaqus ID:

Printed on:

COMPOSITE SHELLS

compshell3_std_sc8r_stack_3.inp Three-layer plate with SC8R elements, three elements


compshell3gs_std_sc8r_stack_3.inp Three-layer plate with SC8R elements, three elements

stacked through the thickness using a general shell section

definition.

compshell3_std_sc8r_stack_6.inp Three-layer plate with SC8R elements, six elements


compshell3_std_sc8r_stack_12.inp Three-layer plate with SC8R elements, 12 elements


compshell3_std_sc8r_stack_24.inp Three-layer plate with SC8R elements, 24 elements


compositeshells_sc8r.f User subroutine DLOAD used with the SC8R models.

Abaqus/Explicit input files

compshell3_1.inp Three-layer plate modeled with S4R elements.

compshell3_1_sc8r.inp Three-layer plate modeled with SC8R elements.

compshell3_1_sc8r_stackdir_1.inp Three-layer plate modeled with SC8R elements using

STACK DIRECTION=1.


STACK DIRECTION=2.


STACK DIRECTION=3.

compshell3_2.inp Three-layer plate with a different thickness and modeled

with S4R elements.

compshell2_1.inp Two-layer plate modeled with S4R elements.

compshell2_2.inp Two-layer plate modeled with S4R elements.

compshell2_1_sc8r.inp Two-layer plate modeled with SC8R elements.

Reference

Pagano, N. J., Exact Solutions for Composite Laminates in Cylindrical Bending, Journal of

Composite Materials, vol. 3, pp. 398411, 1969.

1.1.36

Abaqus ID:

Printed on:

COMPOSITE SHELLS

or

h

x

z

l

p = p sin0 ( )xl

Figure 1.1.31 Composite plate subject to distributed loading.

0 1 2 3

span-to-thickness (*10**1)

0

1

2

3

4

5

w


1 2 Layer: S8R +1.00E+00

2 CPT +1.00E+00

3 Elasticity +1.00E+00

4 3 Layer: S8R +1.00E+00

5 CPT +1.00E+00


1

12

3

3

3

4

45

6

6

6

Figure 1.1.32 Maximum deflection of two- and three-layer plateswith various span-to-thickness ratios; Abaqus/Standard analysis.

1.1.37

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0 1 2 3 4

Transverse Shear/Po

-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

2

2

2

2

3

3

3

3

Figure 1.1.33 Transverse shear stress distribution through thethickness of a two-layer plate ( 4); Abaqus/Standard analysis.

-3 -2 -1 0 1 2 3

Axial Stress/Po (*10**1)

-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

2

2

2

3

3

Figure 1.1.34 Axial stress distribution through the thickness ofa two-layer plate ( 4); Abaqus/Standard analysis.

1.1.38

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0 1 2

Transverse Shear/Po

-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

1

2

2

2

2

2

3

3

3

3

3

3

Figure 1.1.35 Transverse shear stress distribution through thethickness of a three-layer plate ( 4); Abaqus/Standard analysis.

-2 -1 0 1 2


-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

1

2

2

2

2

2

3

3

3

3

3

Figure 1.1.36 Axial stress distribution through the thickness of a three-layerplate ( 4); Abaqus/Standard analysis.

1.1.39

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0.00 0.50 1.00 1.50 2.00

0.50

0.25

0.00

0.25

0.50

Transverse Shear/Po

z/h

ElasticitySSAVG4

-

-

Figure 1.1.37 Comparison of the elasticity solution of the transverse shear stressdistribution in a three-layer plate to the output variable SSAVG4 with 24 SC8R elements

stacked through the thickness; Abaqus/Standard analysis.

0.00 0.50 1.00 1.50 2.00

0.50

0.25

0.00

0.25

0.50

Transverse Shear/Po

z/h

CTSHR13Elasticity

-

-

Figure 1.1.38 Comparison of the elasticity solution of the transverse shear stressdistribution in a three-layer plate to the output variable CTSHR13 with 3 SC8R elements


1.1.310

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.50

0.25

0.00

0.25

0.50

Transverse Shear/Po

z/h

CTSHR13Elasticity

-

-

Figure 1.1.39 Comparison of the elasticity solution of the transverse shear stressdistribution in a two-layer plate to the output variable CTSHR13 with 2 SC8R elements


0 5 10 15 20

Transverse Shear/Po (*10**-1)

-10

-5

0

5

10

z/h

(*10**-1)LINE VARIABLE SCALE FACTOR

1 shell +5.00E-01

2 solid +5.00E-01

1

1

1

2

2

2

Figure 1.1.310 Transverse shear stress distribution throughthe thickness of a three-layer plate ( 4): shells versus solid

elements; Abaqus/Standard analysis.

1.1.311

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0 1 2 3 4 5

Transverse Shear/Po

-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

1

2

2

2

2

2

3

3

3

3

3

Figure 1.1.311 Transverse shear stress distribution through thethickness of a three-layer plate ( 10); Abaqus/Standard analysis.

-1 0 1


-1

0

1

z/h


1 S8R +1.00E+00

2 CPT +1.00E+00


1

1

1

2

2

2

2

2

3

3

3

3

Figure 1.1.312 Axial stress distribution through the thickness of a three-layerplate ( 10); Abaqus/Standard analysis.

1.1.312

Abaqus ID:

Printed on:

COMPOSITE SHELLS

-10 -5 0 5 10

z/h (*10**-1)

0

2

4

6

8

Failure Index


1 Maximum Stress +1.00E+00

2 Tsai-Wu +1.00E+00

3 failure +1.00E+00

1

12

2

3

Figure 1.1.313 Maximum stress theory and Tsai-Wu theory ( 0.0) failure indices as a function ofnormalized distance from the midsurface. Two-layer plate, 4; Abaqus/Standard analysis.

-5 -3 -1 1 3 5

z/h (*10**-1)

0

2

4

6

8

10

Failure Index


1 Maximum Stress +1.00E+00

2 Tsai-Wu +1.00E+001

11

2

2

2

Figure 1.1.314 Maximum stress theory and Tsai-Wu theory ( 0.0) failure indices as a functionof normalized distance from the midsurface. Three-layer plate, 4; Abaqus/Standard analysis.

1.1.313

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Transverse Shear/Po

-1.0

-0.5

0.0

0.5

1.0

z/h

XMIN 0.000E+00XMAX 2.929E+00

YMIN -5.000E-01YMAX 5.000E-01

S4R

CPT

Elasticity

Figure 1.1.315 Transverse shear stress distribution through the thickness of atwo-layer plate; Abaqus/Explicit analysis.

-20. 0. 20.

Axial Stress/Po

-1.0

-0.5

0.0

0.5

1.0

z/h

XMIN -2.739E+01XMAX 2.425E+01

YMIN -5.000E-01YMAX 5.000E-01

S4R

CPT

Elasticity

Figure 1.1.316 Axial stress distribution through the thicknessof a two-layer plate; Abaqus/Explicit analysis.

1.1.314

Abaqus ID:

Printed on:

COMPOSITE SHELLS

0.0 0.5 1.0 1.5 2.0

Transverse Shear/Po

-1.0

-0.5

0.0

0.5

1.0

z/h

XMIN 0.000E+00XMAX 1.768E+00

YMIN -5.000E-01YMAX 5.000E-01

S4R

CPT

Elasticity

Figure 1.1.317 Transverse shear stress distribution through the thickness of athree-layer plate; Abaqus/Explicit analysis.

-20. -15. -10. -5. 0. 5. 10. 15. 20.

Axial Stress/Po

-1.0

-0.5

0.0

0.5

1.0

z/h

XMIN -2.000E+01XMAX 2.000E+01

YMIN -5.000E-01YMAX 5.000E-01

S4R

CPT

Elasticity

Figure 1.1.318 Axial stress distribution through the thicknessof a three-layer plate; Abaqus/Explicit analysis.

1.1.315

Abaqus ID:

Printed on:

COMPOSITE SHELLS

-0.4 -0.2 0.0 0.2 0.4

z/h

0.

1.

2.

3.

4.

5.

6.

Failure Index

XMIN -5.000E-01XMAX 5.000E-01

YMIN 1.354E-01YMAX 6.065E+00

Maximum Stress

Tsai-Wu

FAILURE

Figure 1.1.319 Maximum stress theory and Tsai-Wu theoryfailure indices as a function of normalized distance from the

midsurface. Two-layer plate; Abaqus/Explicit analysis.

-0.4 -0.2 0.0 0.2 0.4

z/h

0.0

0.2

0.4

0.6

0.8

Failure Index

XMIN -5.000E-01XMAX 5.000E-01

YMIN 1.189E-06YMAX 8.285E-01

Maximum Stress

Tsai-Wu

Figure 1.1.320 Maximum stress theory and Tsai-Wu theory failureindices as a function of normalized distance from the midsurface.

Three-layer plate; Abaqus/Explicit analysis.

1.1.316

Abaqus ID:

Printed on:

THICK COMPOSITE CYLINDER

1.1.4 THICK COMPOSITE CYLINDER SUBJECTED TO INTERNAL PRESSURE


This example provides verification of the composite solid (continuum) elements in Abaqus. The problem

consists of an infinitely long composite cylinder, subjected to internal pressure, under plane strain conditions.

The solution is compared with the analytical solution of Lekhnitskii (1968) and with a finite element model

where each layer is discretized with one element through the thickness. A finite element analysis of this

problem also appears in Karan and Sorem (1990).

Most composites are used as structural components. Shell elements are generally recommended to

model such components. Illustrations of composite shell elements in bending can be found in Analysis of

an anisotropic layered plate, Section 1.1.2; Composite shells in cylindrical bending, Section 1.1.3; and

Axisymmetric analysis of bolted pipe flange connections, Section 1.1.1 of the Abaqus Example Problems

Manual. In some cases, however, the analyst cannot avoid the use of continuum elements to model structural

components. In these problems careful selection of the element type is usually essential to obtain an accurate

solution. The performance of continuum elements for the analysis of bending problems is discussed in

Performance of continuum and shell elements for linear analysis of bending problems, Section 2.3.5.

The discussion considers only the behavior of structures composed of homogeneous materials, but the

same considerations apply when modeling composite structures with continuum elements. In other cases

the deformation through the thickness of the composite may be nonlinearfor example, when material

nonlinearities are presentand several elements may be required through the thickness for an accurate

analysis. Such a discretization can be accomplished only with continuum elements. Other problems where

the use of continuum elements may be preferred include thick composites where transverse shear effects are

predominant, composites where the normal strain cannot be ignored, and when accurate interlaminar stresses

are required; i.e., near localized regions of complex loading or geometry. In these problems the solutions

obtained by solid elements are generally more accurate than those obtained by shell elements. An exception

is the distribution of transverse shear stress through the thickness. The transverse shear stresses in solid

elements usually do not vanish at the free surfaces of the structure and are usually discontinuous at layer

interfaces. A discussion of the transverse shear stress calculations for solid and shell elements can be found

in Composite shells in cylindrical bending, Section 1.1.3.

In this problem the normal strain cannot be ignored since the displacement field due to the internal

pressure is nonlinear through the cylinder thickness. At least two quadratic elements through the thickness are

required to obtain accurate results. The example, therefore, demonstrates the use of composite solid elements

for a problem where a shell element analysis would be inadequate.

Problem description

The cylinder configuration and material details are shown in Figure 1.1.41. The inside radius, , is

60 mm, and the outside radius, , is 140 mm. The structure consists of eight orthotropic layers of

equal thickness, arranged in a stacking sequence of [0, 90]4 . The laminae are stacked in the radial

direction, with the material fibers oriented along the circumferential and axial directions. In other words,

the fibers are rotated 0 or 90 about the radial direction, where a 0 rotation implies primary fibers

1.1.41

Abaqus ID:

Printed on:


oriented along the circumferential direction. For this purpose we define a local coordinate system using

the *ORIENTATION option, where the 1, 2, and 3 directions refer to the radial, circumferential, and axial

directions, respectively. The fiber composite with the primary fibers along the circumferential direction

has the following orthotropic elastic properties in this coordinate system:

10.0 GPa, 250.0 GPa, 10.0 GPa,

5.0 GPa, 2.0 GPa,

0.01, 0.25.

We also define the composite with the primary fibers along the axial direction of this local coordinate

system. Recognizing that the Poissons ratios, , must obey the relations for an orthotropic

material with engineering constants, the rotated material properties are

10.0 GPa, 10.0 GPa, 250.0 GPa,

2.0 GPa, 5.0 GPa,

0.25, 0.01.

Each of these sets of material properties is specified on the *ELASTIC, TYPE=ENGINEERING

CONSTANTS option. The name of each material is referred to on the data lines following the *SOLID

SECTION, COMPOSITE option. This material definition ensures that the output components in the

different layers are provided in the same coordinate system.

There is another method in Abaqus that can be used to define the ply orientation of the composite

material. In this method only one definition of the material properties is used, but a separate orientation

definition is given for each layer. This layer orientation is specified, together with the material name, on

the data lines following the *SOLID SECTION option. The orientation can be specified by referring to an

*ORIENTATION definition or by specifying an angle relative to the section orientation definition. The

section orientation is specified with the ORIENTATION parameter on the *SOLID SECTION option.

Since the material properties of each layer in this case are specified in a different local coordinate system,

the output variables are provided in different coordinate systems. Input files illustrating both methods

are provided.

In addition to the material description for each layer, we need to define the stacking direction, the

thickness of each layer, and the number of section points through the layer thickness required for the

numerical integration of the element matrices to complete the description of the composite arrangement.

The stacking direction is specified on the *SOLID SECTION option with the STACK DIRECTION

parameter, and the thickness and number of integration points are specified on the data lines following the

*SOLID SECTION option. Three section integration points are specified in each layer. Since the analysis

is linear elastic, this is sufficient to describe the stress distributions through the section. The layers can be

stacked in any of the three isoparametric element coordinate directions, whichin turnare defined by

the order in which the nodes are given on the element data line. In this example the element connectivity

is specified so that the first isoparametric direction lies along the radial direction.

1.1.42

Abaqus ID:

Printed on:


Geometry and model

Because of symmetry, only a segment of the body needs to be analyzed. For simplicity of boundary

condition application a quarter segment is chosen and is discretized with four elements in the

circumferential direction and one element in the axial direction. One, two, four, or eight elements are

used in the radial direction. Figure 1.1.42 shows the finite element discretization for the case where

two elements are used in the radial direction. A nonuniform mesh, with two material layers in the inside

element and six layers in the outside element, is used to capture the variation of the radial displacement

through the section.

The model is bounded in the axial direction to impose plane strain conditions.

The load is a constant internal pressure of 50 MPa applied in a linear perturbation step.


All displacements and stresses reported here are normalized with respect to pressure, using

The predicted displacements and stresses at the inside and outside surfaces of the cylinder are

compared with the analytical results in Table 1.1.41 and Table 1.1.42. Results are shown for different

element types and for different mesh densities. The tables show that a model discretized with one solid

element (linear or quadratic) in the radial direction is inadequate to model the nonlinear variation of the

displacement field. A substantial improvement is obtained with two elements through the thickness. The

tables further show that the convergence of the finite element results onto the analytical solution is slow

with mesh refinement. A mesh with two nonuniform quadratic elements through the thickness predicts

remarkably accurate results, with the exception of the circumferential stress at the outside surface of the

cylinder. The outside stress is, however, more than two orders of magnitude smaller than the inside stress

and is, therefore, not a good measure of the accuracy of the solution.

The displacement and stress fields through the thickness are shown in Figure 1.1.43 through

Figure 1.1.45. The figures compare the normalized radial displacement, the circumferential stress,

and the radial stress with the analytical solution for the case where the cylinder is discretized with

two C3D20R elements (of different sizes) in the radial direction. The figures show that the radial

displacement and circumferential stress are in good agreement with the analytical solution. The radial

stress, especially near the inside of the cylinder, is not quite as accurate. For example, the analytical

solution at the inside surface is 1.0 ( ). The finite element result for this mesh is

0.741 (25.9% error). This result must be seen in light of mesh refinement; no improvement in

the radial stress at the inside surface is obtained with four elements through the thickness, and it only

improves to 0.926 (7.4% error) when eight elements are used through the thickness (the results

for the four-element and eight-element meshes are not shown in the figures). It is clear from these

figures why quadratic elements and a refined mesh are required for an accurate analysis.

1.1.43

Abaqus ID:

Printed on:


Input files

thickcompcyl_2el_nonuniform.inp Model discretized with two nonuniform elements in the

radial direction.

thickcompcyl_1el_sectorient.inp Model in which the ply orientation is specified with a

rotation relative to the section orientation. This model is

discretized with one element in the radial direction.

thickcompcyl_4el_orient.inp Model in which the ply orientation is specified with an

orientation reference. This model is discretized with four

elements in the radial direction.

thickcompcyl_8el.inp Model in which each layer is discretized with one

homogeneous element through the thickness.

References

Karan, S. S., and R. M. Sorem, Curved Shell Elements Based on Hierarchical p-Approximation in

the Thickness Direction for Linear Static Analysis of Laminated Composites, International Journal

for Numerical Methods in Engineering, vol. 29, pp. 13911420, 1990.

Lekhnitskii, S. G., Anisotropic Plates, translated from second Russian edition by S. W. Tsai and T.

Cheron, Gordon and Breach, New York, 1968.

1.1.44

Abaqus ID:

Printed on:


Table 1.1.41 Normalized radial displacement at inside and outside ofcylinder. Analytical solution: 1.4410; 0.1476.

Inside OutsideElementtype

Elements in radialdirection % error % error

C3D8 1 1.1825 17.9 0.2407 263.0

C3DI 1 1.2227 15.2 0.1004 32.0

C3DI(1) 2 1.4231 12.4 0.1876 27.1

C3DI(2) 2 1.5526 7.74 0.1828 23.8

C3D20R 1 1.2581 12.7 0.1646 11.5

C3D20R(1) 2 1.3609 5.56 0.1448 1.90

C3D20R(2) 2 1.3869 3.75 0.1481 0.34

C3D20R 4 1.3922 3.39 0.1447 1.95

C3D20R 8 1.4161 1.73 0.1496 1.35

1 - Uniform mesh

2 - Nonuniform mesh

Table 1.1.42 Normalized circumferential stress at inside and outsideof cylinder. Analytical solution: 5.7060; 0.0103.

Inside OutsideElementtype

Elements in radialdirection % error % error

C3D8 1 3.608 36.8 0.0307 397.0

C3DI 1 3.912 31.4 0.0362 251.1

C3DI(1) 2 4.686 17.9 0.004 60.8

C3DI(2) 2 4.838 15.2 0.0081 179.1

C3D20R 1 5.132 10.1 0.0414 300.0

C3D20R(1) 2 5.496 3.68 0.0134 30.0

C3D20R(2) 2 5.548 2.77 0.0192 85.6

C3D20R 4 5.574 2.31 0.0119 15.1

C3D20R 8 5.606 1.75 0.0107 3.90

1 - Uniform mesh

2 - Nonuniform mesh

1.1.45

Abaqus ID:

Printed on:


d

d

y

x

Lamina 8: 90Lamina 7: 0Lamina 6: 90Lamina 5: 0Lamina 4: 90Lamina 3: 0Lamina 2: 90Lamina 1: 0

o

o

o

o

o

o

o

o

t

P

centerline

i

Po

Figure 1.1.41 Geometry of laminated cylinder.

1

2

3 1

2

3

Figure 1.1.42 Finite element discretization with two elements in the radial direction.

1.1.46

Abaqus ID:

Printed on:


6 8 10 12 14

Radial direction (*10**1)

0

5

10

15

Normalized displacement


1 analytical +2.00E+01

1

1

1

1

1

11

1 1 1

6 8 10 12 14

(*10**1)

0

5

10

15


1 analytical +2.00E+01

2 2 element +2.00E+01

12

Figure 1.1.43 Radial displacement versus cylinder radius.

6 8 10 12 14


0

1

2

3

4

5

6

Normalized Stress


1 analytical +2.00E-02

1

1

1

1

1

1

1 1

1

16 8 10 12 14

(*10**1)

0

1

2

3

4

5

6



2 2 element +2.00E-02

12

2 2 2

2

Figure 1.1.44 Circumferential stress versus cylinder radius.

1.1.47

Abaqus ID:

Printed on:


6 8 10 12 14


-10

-8

-6

-4

-2

0Normalized Stress



1

11

1

11

1 11 1

6 8 10 12 14

(*10**1)

-10

-8

-6

-4

-2

0



2 2 element +2.00E-02

1

2

2 2

2

2

Figure 1.1.45 Radial stress versus cylinder radius.

1.1.48

Abaqus ID:

Printed on:

UNIFORM COLLAPSE OF PIPE

1.1.5 UNIFORM COLLAPSE OF STRAIGHT AND CURVED PIPE SEGMENTS


The failure of pipe segments under conditions of pure bending is an interesting problem of nonlinear structural

response. In the case of straight, thin-walled, metal cylinders, the failure usually occurs by the cylinder

buckling into a pattern of small, diamond-shaped waves, in the same fashion as a cylinder failing under axial

compression

abaqus benchmarks manual

Documents